2021/10/21 by Christopher F. Dowd, Dowd, Christopher · 1 citation
Computer Science · Mathematics · Social Sciences · #Algebraic Geometry and Number Theory #Cryptography and Residue Arithmetic #FOS: Mathematics #Historical and Political Studies #Number Theory (math.NT) #Polynomial and algebraic computation
paper · pdf · doi:10.48550/arxiv.2110.13908
openalex publication_date 2021/10/21 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Elliptic K-curves are elliptic curves defined over some field extension\nL/K that are isogenous to all of their Galois conjugates. We present a new\nresult on K-curves E that are given by a K-rational orbit \τ,\n-1/N\τ of the Fricke involution on Y0(N), giving a simple Diophantine\ncondition on the extension L/K that determines which twists of E allow the\nisogeny between Galois conjugates to be defined over L. To support and\nillustrate this result, we also discuss parameterizations of cyclic\nN-isogenies corresponding to points on modular curves X0(N) of genus 0.\nThese modular curves admit parameterizations in terms of a distinguished\nHauptmodul. We provide an exposition on the derivation of these Hauptmoduln as\nproducts of the Dedekind eta function based on the approach of Ligozat. As an\napplication, we provide a complete tabulation of explicit formulas for the\ncoefficients of cyclic N-isogenous curves in terms of the Hauptmodul for all\nN such that X0(N) has genus 0. We also include an abbreviated table of\nrational functions for the j-invariant in terms of Hauptmoduln, and we\ndiscuss a classical application of these expressions to finding special values\nof the j-invariant at CM points.\n